Chapter 5

FUZZY NUMBER

Chi-Yuan Yeh

2

Outline

• Interval and Fuzzy number• Operations of interval• Operations of fuzzy number

– α-cut

– extension principle

• Triangular fuzzy number• Trapezoidal fuzzy number

Interval

31a 3a

1 3[ , ]a a

α-cut

4

( )1a ( )

3a

( ) ( )1 3,A a a

( ) ( )1 3

1

,A

a a

α-cut

5

( )1a ( )

3a

( ) ( )1 3,

Aa a

[0,1]A A

α-cut

6

Fuzzy number

• Definition: a fuzzy set A on R must posses at least the following three properties– A must be a normal fuzzy set

– must be a closed interval for every (convex)

– the support of fuzzy set A, , must be bounded

7

( , 1]o A

0A

Operations of Interval

8

Operations of Interval

9

Operations of Interval

10

Operations of Interval

11

Operations of Interval

12

Operations of Fuzzy Number

13

• α-cut– Addition:

[0,1]( ) ( ( ) )A B A B

Operations of Fuzzy Number

14

• α-cut– Subtraction:

– Multiplication:

– Division:

– Minimum:

– Maximum:

[0,1]( ) ( ( ) )A B A B

[0,1]( ) ( ( ) )A B A B

[0,1]( ) ( ( ) )A B A B

[0,1]( ) ( ( ) )A B A B

[0,1](/) ( (/) )A B A B

Operations of Fuzzy Number

15

Operations of Fuzzy Number

• α-cut

• α = 0.5– –

• α = 1.0– –

•

16

0.5 0.5[2,3], [3, 4]A B

0.5 0.5( ) [5,7]A B

1.0 1.02, 3A B

1.0 1.0( ) 5A B

0.5 0.5 1.0 1.0

0.5 1.0( ) 0.5( ( ) ) 1.0( ( ) )

[5,7] 5A B A B A B

1.0 0.5 0.5( )

5 6 7A B

Operations of Fuzzy Number

17

• Extension principle

– Addition:

– Subtraction:

– Multiplication:

– Division:

– Minimum:

– Maximum:

BA )( ))()(()()( yxz BAyxz

BA

BA )( ))()(()()( yxz BAyxz

BA

BA )( ))()(()()( yxz BAyxz

BA

zyx ,,

BA(/) ))()(()(/

(/) yxz BAyxz

BA

BA )( ))()(()()( yxz BAyxz

BA

BA )( ))()(()()( yxz BAyxz

BA

Operations of Fuzzy Number

18

Operations of Fuzzy Number

19

Operations of Fuzzy Number

20

Operations of Fuzzy Number

21

Operations of Fuzzy Number

22

Operations of Fuzzy Number

23

Operations of Fuzzy Number

24

Operations of Fuzzy Number

25

Triangular fuzzy number

26

Triangular fuzzy number

27

Triangular fuzzy number

28

Triangular fuzzy number

29

Operations of triangular fuzzy number

30

Operations of triangular fuzzy number

31

Operations of triangular fuzzy number

32

Operations of triangular fuzzy number

33

Operations of triangular fuzzy number

34

Operations of triangular fuzzy number

35

Operations of triangular fuzzy number

36

Operations of triangular fuzzy number

37

Operations of triangular fuzzy number

38

Operations of triangular fuzzy number

39

Operations of triangular fuzzy number

40

Operations of triangular fuzzy number

41

Operations of triangular fuzzy number

42

Operations of triangular fuzzy number

43

2

2

2 4 2

2 2

2

2

z

z

z

2

2

4 20 24

2 5 1

1 5

2

z

z

z

( ( ) )

0 2

22 8

2( )1 5

8 2420 24

A B

z

zz

zz

z

z

Approximation of Triangular Fuzzy Number

44

Approximation of Triangular Fuzzy Number

45

46

Approximation of Triangular Fuzzy Number

Operations of triangular fuzzy number

47

Operations of triangular fuzzy number

48

1

2 62 1 6 1

6 1

2 1

z

z z

z

z

2 4

2 22 2 4 2

4 2

2 2

z

z z

z

z

( (/) )

10

66 1 1 1

( ) 2 1 6 24 2 1

22 2 2

0 2

A B

z

zz

z zz

zz

z

4

Approximation of Triangular Fuzzy Number

49

4

Approximation of Triangular Fuzzy Number

50

( (/) )

0 0.17

0.70.17 0.5

0.33( )2

0.5 21.50 2

A B

z

zz

zz

z

z

Trapezoidal fuzzy number

51

Operations of Trapezoidal Fuzzy Number

52

Operations of Trapezoidal Fuzzy Number

53

54

2

2

4 9 2

2 2.25 3.0625

3.0625 2.25

2

z

z

z

2

2

9 51 72

3 8.5 0.25

0.25 8.5

3

z

z

z

( ( ) )

0 2

3.0625 2.252 15

2( ) 1 15 30

0.25 8.530 72

30 72

A B

z

zz

z z

zz

z

Operations of Trapezoidal Fuzzy Number

Approximation of Trapezoidal Fuzzy Number

55

( ( ) )

0 2

22 15

13( ) 1 15 30

7230 72

420 72

A B

z

zz

z z

zz

z

Operations of Trapezoidal Fuzzy Number

56

Exercise

57

0 3

33 5

2( )10

5 1050 10

B

x

xx

xx

x

x

Is B a convex fuzzy set?

Is B a fuzzy number?

Exercise

58

59

Thanks for your attention!